Let $f \in H^{-1}(\Omega)$. We consider the problem
$$
\begin{cases}
& - \mathrm{div} A(x,u^{\epsilon}) \nabla u^{\epsilon} + g_{\epsilon} (x,u^{\epsilon})=f\\
& u^{\epsilon} \in H^1_0(\Omega)
\end{cases}
$$
How we can prove with the Schauder fixed point theorem that this problem admits one last solution $u_{\epsilon} \in H^1_0(\Omega)$?
Have you an book or a paper? Thanks for the help.